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1.63 Use the displacement form of the governing equations to compute the torsional con-
stant and stresses for a uniform bar of elliptical cross-section (Fig. 1.19). Assume the
warping function is of the form
ω
=
cyz,
where
c
is a constant.
Hint:
From Eq. (1.145),
dy
ds
τ
dz
ds
τ
a
z
τ
+
a
y
τ
=−
+
=
0
(1)
xz
xy
xz
xy
Introduce
xy
of Eq. (1.166). Calculate the direction cosines from the equation
of an ellipse (Fig. 1.19) and find
τ
xz
and
τ
a
2
z
z
b
2
y
+
∂ω
∂
+
∂ω
∂
−
y
+
=
0
z
y
Substitute
ω
=
cyz
, which satisfies Laplace's equation, into this expression and solve
for
c
.
Alternatively, substitute
dy
ds
y
dz
ω
=
cyz
into (1) to find
(
−
1
+
c
)
−
ds
z
(
1
+
c
)
=
0
.
constant. Comparison with
y
2
b
2
a
2
Integrate, giving
y
2
(
1
−
c
)
(
z
2
z
2
b
2
)
+
=
+
=
for the
1
+
c
ellipse of Fig. 1.19 leads to the value of
c
.
Answer:
a
2
−
b
2
a
3
b
3
π
=
=
c
b
2
,
J
a
2
a
2
b
2
+
+
2
ab
3
2
a
3
b
τ
xy
=−
M
t
z,
τ
xz
=
M
t
y
π
π
1.64 Compare the torsion of a shaft of elliptical cross-section with that of a shaft of circular
cross-section with a radius equal to the minor axis
b
of the ellipse. Show that for
the same angle of twist the maximum shear stress will be greater in the elliptical
cross-section.
1.65 A bar subject to torsion has a cross-section in the shape of an equilateral triangle as
shown in Fig. P1.65. Compute the maximum shear stress. Also find the angle of twist
per unit length, the torsional constant
J,
and the torsional rigidity
GJ
.
Begin with the
stress function
y
y
y
√
3
z
√
3
z
φ
2
h
G
2
h
3
2
h
3
h
3
ψ
=
−
−
+
−
+
Answer:
15
√
3
M
t
2
h
3
τ
=
max
h
4
15
√
3
G
h
4
15
√
3
J
=
M
x
=
θ
1.66 Find the warping function for a prismatic bar with a rectangular-shaped cross-section
(Fig. P1.66). Begin with a warping function of the form
ω
=
yz
+
c
sin
kz
sinh
ky
where
c
and
k
are constants to be identified such that the boundary conditions are
satisfied.
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